Some conceptual understanding problems. For instance I know that if $X$ is path connected, then $X$ is connected, but it is not necessarily true the other way around. In the example of my book, they give some set $X$ saying:
For example, when $X= \{(x,\sin\frac{1}{x}):0 <x\le 1\} \subset \mathbb{R}^2_{\text{usual topology}}$, it is clear that $\overline{X}=X\cup [0]\times [-1,1]$ is connected but not path connected.
Now I have hard time seeing why this holds, I would appreciate some clear explanation.